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  • 标题:Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom
  • 作者:Cyril Cohen ; Thierry Coquand ; Simon Huber
  • 期刊名称:LIPIcs : Leibniz International Proceedings in Informatics
  • 电子版ISSN:1868-8969
  • 出版年度:2018
  • 卷号:69
  • 页码:5:1-5:34
  • DOI:10.4230/LIPIcs.TYPES.2015.5
  • 出版社:Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
  • 摘要:This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.
  • 关键词:univalence axiom; dependent type theory; cubical sets
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